Logarithm Calculator

How to use the logarithm calculator

1. Pick a function: log₁₀ (common), ln (natural), log base b or Antilog (bʸ).
2. Type Number x: a whole number, a decimal, a fraction such as 1/8, scientific notation such as 2.5e-7, or the letter e.
3. For log base b and Antilog (bʸ), enter Base b: any positive number other than 1, or e. For the antilog, the first field becomes Exponent y.
4. The result appears as you type (Calculate does the same), and Copy copies it.

For logarithms, the tool also shows log₁₀ x, ln x and log₂ x side by side, and the Step by step panel explains how the result was found.

What a logarithm is

The logarithm answers the question “to what power must I raise the base to get this number?”. log_b(x) = y means exactly bʸ = x. So log₁₀(1000) = 3 because 10³ = 1000, and log₂(8) = 3 because 2³ = 8.

• The common logarithm (log or log₁₀) uses base 10. It counts orders of magnitude and is behind the pH scale, decibels and the Richter scale.
• The natural logarithm (ln) uses the number e ≈ 2.718281828. It appears in growth, decay, compound interest and calculus, because the derivative of ln x is 1/x.
• The binary logarithm (log₂) counts doublings; it is used in computing and information theory.
• The antilog reverses a logarithm: the antilog of y in base b is bʸ. The antilog of 2.5 in base 10 is 10^2.5 ≈ 316.23.

Exact answers when they exist

Many logarithms are rational numbers, and this tool finds them instead of printing a rounded decimal. It writes both numbers as powers of a common base: 8 = 2³ and 4 = 2², so log₄(8) = 3/2. It also handles fractions (log₄(1/2) = −1/2) and negative results (log₁₀(0.001) = −3). When no exact form exists, such as log₃(10), the answer is irrational and is given to about 14 significant digits.

The change-of-base formula

Most calculators only have log and ln keys. Any other base goes through the change-of-base formula:

log_b(x) = ln(x) / ln(b) = log(x) / log(b)

For example, log₃(10) = ln 10 / ln 3 = 2.302585 / 1.098612 ≈ 2.0959. The steps show this division with both natural and common logarithms, then check the answer by raising the base to the result.

For base 10 there is another useful step: write x in scientific notation, m × 10ᵏ, then log₁₀(x) = k + log₁₀(m). For 12,345 = 1.2345 × 10⁴ this gives 4 + 0.0915 = 4.0915. That is also how the tool handles numbers far beyond ordinary calculator range, such as 10⁵⁰⁰.

Rules worth remembering

• log_b(xy) = log_b(x) + log_b(y)
• log_b(x / y) = log_b(x) − log_b(y)
• log_b(xⁿ) = n · log_b(x)
• log_b(1) = 0 and log_b(b) = 1